### Kaluza-Klein GUTs in String Theory

M. Wijnholt, LMU

Taiwan, Dec 2010

w/R.Donagi, T.Pantev, L.Anderson

Plan:

* General philosophy

* The type I’ story

* Degenerating Branes and F-theory

Part I: General Philosophy

We can construct 4d GUT models by compactifying a higher dimensional supersymmetric gauge theory with gauge group G.

To get avoid certain problematic interactions, we use exceptional gauge groups.

We further want to be able to embed our KK GUTs in string theory, as higher dimensional gauge theories are highly non-renormalisable.

This works beautifully in the heterotic string, where we can start with an E8 x E8 gauge group in 10d.

Now we imagine we are a little bored with the heterotic string, and we want to try GUT models with grow fewer extra dimensions at the GUT scale.

Thus although we may be still able to embed such models in the big M- theory moduli space, string perturbative techniques cannot be used; we need another approximation scheme and another set of methods.

In fact, even in the heterotic string, one often does not use string perturbation theory, as this can only be used when the worldsheet theory is free or exactly solvable.

Instead, one typically uses 1/g_4^2 = V/g_YM^2 as the small

parameter, I.e. one simply uses the 10d Yang-Mills Lagrangian. The stringy corrections can essentially be ignored in this limit.

This can be (partially) generalized any dimension.

Except for heterotic, exceptional gauge groups can not be obtained in perturbative string theory.

However, since the field theory is lower dimensional, clearly we cannot use this approach globally.

Thus we proposed to split the program into two parts:

Local model

Compactified Yang-Mills, aka Higgs bundles

Global model

Special holonomy manifold with ADE singularities

To embed local in global, we represent the Higgs bundle by ALE fibration and paste it in.

10d -- Heterotic

9d –- type I’

8d –- F-theory

7d –- M-theory

Let us take stock of KK GUT models in string theory

Donagi/MW, 08

Beasley/Heckman/Vafa, 08

Pantev/MW, 09 Candelas et al, 85

And to appear

?

Hayashi et al, 08

10d SYM on CY_3, gauge field connection on E_8 bundle V:

### ε δλ =

_{μν}

### Γ

^{μν}

### = *F*

### 0

### = 0

*j*
*i*
*j*
*i*

*g* 0 *F*

2 ,

0

### =

*F*

Massless fields from KK reduction of E_8 gauginos:

### = 0 /

_{A}### λ

*D*

Fermions on Z (0,i) forms

Dirac operator Dolbeault operator

Massless gauginos:

Massless chiral fields:

### ) ,

### (

80

*V*

*E*

*Z* *H*

### ) ,

### (

81

*V*

*E*

*Z* *H*

Heterotic story:

Yukawas: ^{1}

### ( , )

^{3}

### C

8 ^{⊗}

### →

*V*

*E*

*Z*

*H*

Analogue in lower dimensions: (parabolic) Higgs bundles

* Bundle E with connection

* Adjoint field , interpreted as a map

*A*

μ
### Φ *E* → *E* ⊗ *N*

This data has to satisfy first order BPS equations

Hitchin’s equations

F-theory story:

8d SYM is dimensional reduction of 10d SYM:

0 , 2 1

, 0 1

,

0

*= A* + Φ

### A

8d SYM on compact Kaehler surface S:

E_8 bundle V

Higgs field Φ ^{•}

### = *V*

_{E}### ⎯ ⎯→

^{Φ}

*V*

_{E}### ⊗ *K*

_{S}8

### E

8### = 0

### δλ _{F}

_{F}

^{0}

^{,}

^{2}

_{=} _{0} _{F}

_{F}

_{i}

_{j}_{g}

_{g}

^{i}

^{j}_{+} _{[} _{Φ} _{,} _{Φ}

^{*}

_{]} _{=} _{0}

Hitchin
Massless gauginos:

Massless chiral fields:

### ) ,

0

### (

•### E *S* *H*

### ) ,

1

### (

•### E *S* *H*

### = 0 Φ

### ∂

_{A}^{,}

: ,

M-theory story:

7d SYM is dimensional reduction of 10d SYM:

### φ *i* *A* + A =

7d SYM on compact real 3-manifold Q:

E_8 bundle V

Higgs field φ

### E

^{•}

### = *V* ⎯ ⎯→

^{φ}

*V* ⊗ *T*

^{*}

*Q*

### = 0

### δλ ^{F} ^{+} ^{[} ^{φ} ^{,} ^{φ} ^{]} ^{=} ^{0} *d*

^{F}

^{*}

_{A}### φ ^{=} ^{0}

_{Hitchin}

Massless gauginos:

Massless chiral fields:

### ) ,

0

### (

•### E *Q* *H*

### ) ,

1

### (

•### E *Q* *H*

### = 0

*A*

### φ

*d*

^{,}

: ,

Part II: The type I’ story

Usual description: S^1/Z_2 with piecewise linear function H \sim e^-\phi

This description is somewhat clumsy at strong coupling

When the string coupling diverges at the end, can get exceptional gauge symmetries. [Polch/Witten]

x_{9}
H

Alternative description:

Elliptic K3 surfaces with a real involution. Cachazo/Vafa

This yields the picture most familiar from M/F-theory:

Enhanced gauge symmetry ADE singularities Abelian gauge fields Real C_3

*g* *fx*

*x*

*y*

^{2}

### =

^{3}

### + +

Suppose we want an SU(5) gauge symmetry. Then K3 takes the form:

*xy* *b* *zx*

*b* *y*

*z* *b* *x*

*z* *b* *z*

*b* *x*

*y*

^{2}

### =

^{3}

### +

_{0}

^{5}

### +

_{2}

^{3}

### +

_{3}

^{2}

### +

_{4}

^{2}

### +

_{5}

+ higher order As usual except variables are real

Throw out higher order terms to get local model

Real E_8 ALE unfolded to A_4 singularity

This should be equivalent to a 5d Higgs bundle (Compactified 9d Yang-Mills)

Compactify 9d super Yang-Mills on X_5

X_5 should likely be Einstein-Sasaki

5d version of Hitchin’s equations:

Choose local coordinates z_{1}, z_{2}, x

*w*

### ϕ

*w*
*z*
*x*

*z*

*J* *D*

*F* =

### = 0 +

_{x}### ϕ

*z*
*z*
*z*

*z*

*F* *iD*

*g*

Fields: 5d vector

Real adjoint scalar

*A*

μ
### ϕ

Choose a gauge where

*A*

_{x}### = *A*

_{z}### = 0 , *A*

_{z}### ≠ 0

### = 0

### ∂ ϕ

Spectral cover description with coisotropic 8-branes in Total( )

But not coisotropic in Kapustin-Yi sense.

Examples:

X_5 = unit circle bundle in ( )

Fourier-Mukai transform along S^1

F-theory spectral covers in ( ) with zero section deleted Problem: lift adjoint matter.

*dP* *K* →

*dP* *K* →

*X*

5
*R* →

Part III: Degenerating branes

Start with a simple question: how do we specify an intersecting brane configuration?

1 1

*, L* *D*

2 2

*, L* *D*

### Σ

One also has to specify the gluing morphism on the intersection:

Note this is asymmetric in 1 & 2, and explicitly breaks a U(1) symmetry.

There are a number of equivalent ways to say this.

### ) ,

### ( *L*

_{1}

*L*

_{2}

*Hom*

*f* ∈

_{Σ}

Seems reasonable, but it is incomplete !

1 1

*, L* *D*

2 2

*, L* *D*

### Σ *f*

DW, ‘10

Suppose the intersecting branes are given by an equation

Over the intersection, this is the equation of a non-reduced scheme

There are two natural sheaves over this:

* Rank two sheaf over

* Structure sheaf of , which restricts to a rank one sheaf on

The first corresponds to zero gluing VEV, and the second to non-zero VEV.

### 0 )

### )(

### ( *z* − λ *z* + λ =

### 0 ,

### 0

^{2}

### =

### = λ *z*

2

### = 0 λ

### = 0 λ

### = 0

### λ

Higgs bundle perspective:

### ⎟⎟ ⎠

### ⎜⎜ ⎞

### ⎝

### ⎛

### ≈ −

### Φ *z*

*f* *z*

### 0

### 0 )

### )(

### ( )

### det( λ *I* − Φ ≈ λ − *z* λ + *z* =

Spectral cover equation:

Identify f with the gluing VEV.

This has a lot of applications. Focus on F-theory and the heterotic string.

* Evade no-go theorems

Eg. [BHV2]: SO(10) GUT models have exotics.

This assumed vanishing of the gluing VEV

* Various possibilities for flavour structures Eg. Gluing morphism is asymmetric

Leads to Yukawa textures And/or models with bulk chiral matter

Texture from D-terms

* Understand F-theory duals of heterotic linear sigma models

Spectral cover is often degenerate, so understanding the spectral sheaf is crucial.

Bershadsky et al gave a simple algorithm for computing the spectral cover of a monad.

We generalized algorithm to also get spectral sheaf

### 0 0 → π

^{*}

### π

_{*}

*V* → *V* → *L* →

Many nice results on heterotic linear sigma models, which can now be compared with dual F-theory.

Also gives checks on our claims on degenerate covers.

Finally, although somewhat disconnected from previous discussion, I would like to propose a numerical approach to finding solutions of the D- terms for Higgs bundles.

Review of numerical approach to Hermitian Yang-Mills:

Consider positive L, and sections

### )

0

### (

*m*

*i*

*H* *V* *L*

*s* ∈ ⊗

^{m}

^{>> 0}

Fixed point g=h is called balanced embedding.

### |

2### | log *s*

_{h}*K* =

## ∫

^{−}

### =

^{K}

_{i}^{*}

_{j}*ij*

*e* *s* *s*

*g*

Balanced metric (pull-back of Fubini-Study) converges to the HYM metric.

Conjecture:

For Higgs bundles, same story except replace

*V* → E

^{•}

Consider

Oversimplified, r=1.

Conclusions:

* General story in M/F/type I’: Higgs bundles

* Guts from type I’, but much remains to be explored

* F-theory and heterotic are best developed, due to techniques from algebraic geometry.

* Better understanding of degenerate configurations, D-terms.